Some Prominent Approaches to the Representation of Uncertain Inferences

The following figure indicates some relationships among six of the
most prominent approaches. The arrows point from more general to less
general representation schemes. For example, the
Dempster-Shafer represention contains the probability
functions as a special case.

Representations of Uncertainty

These representations are often described as measures on events, or
states, or propositions, or sets of possibilities. But deductive logics
are usually described in terms of statements or sentences of a
language. So let's follow suit.

Plausibility relations (Friedman and Halpern, 1995)
constitute the most general of these representations. They satisfy the
weakest axioms, the weakest constraints on the logic of uncertainty. For a
plausibility relation ⊆ between sentences, an expression
‘A ⊆ B’,
says that A is no more plausible than B
(i.e., B is at least as plausible as A, maybe more
plausible). The axioms for plausibility relations say that tautologies
are more plausible than contradictions, any two logically equivalent
sentences are plausibility-related to other sentence in precisely the
same way, a sentence is no more plausible than the sentences it
logically entails, and the no more plausible than relation is
transitive. These axioms make plausibility relations weak partial
orderings on the relative plausibility of sentences. They permit
some sentences to be incomparable—neither more plausible, nor
less plausible, nor equally plausible to one another.

Qualitative probability relations are plausibility
relations for which the ordering is total—i.e.
any two sentences are either equally plausible, or one is more
plausible than the other. This total ordering is established
by one additional axiom. Qualitative probability relations
also satisfy a second additional axiom that says that when a sentence
S is logically incompatible with A and with
B, then
A ⊆ B
holds just in case(A or S) ⊆ (B or S)
holds as well. When qualitative probability relations are
defined on a language with a rich enough vocabulary and satisfy one
additional axiom, they can be shown to be representable by
probability functions—i.e., given any qualitative
probability relation ⊆, there is a unique probability
function P such that
A ⊆ B
just in case
P[A]
≤
P[B].
So quantitative probability may be viewed as essentially
just a way of placing a numerical measure on sentences that uniquely
emulates the is no more plausible than relation specfied by
qualitative probability. (See (Koopman, 1940), (Savage,
1954), (Hawthorne and Bovens, 1999), (Hawthorne, 2009).)

Probability (i.e., quantitative probability) is a
measure of plausibility that assigns a number between 0 and 1
to each sentence. Intuitively, the probability of a sentence
S,
P[S]
= r, says that Sis plausible to degree r,
or that the rational degree of confidence (or belief) thatSis true isr. The axioms for
probabilities basically require two things. First,
tautologies get probability 1. Second, when A and B
contradict each other, the probability of the disjunction (A
or B) must be the sum of the probabilities of A and
of B individually. It is primarily in regard to this second
axiom that probability differs from each of the other
quantitative measures of uncertainty.

Like probability, Dempster-Shafer belief functions
(Shafer, 1976, 1990) measure appropriate belief strengths on
a scale between 0 and 1, with contradictions and tautologies at the
respective extremes. But whereas the probability of a
disjunction of incompatible claims must equal the sum of the parts,
Dempster-Shafer belief functions only require such
disjunctions be believedat least as strongly as the
sum of the belief strengths of the parts. So these functions
are a generalization of probability. By simply tightening up
the Dempster-Shafer axiom about how disjunctions are related
to their parts we get back a restricted class of
Dempster-Shaferfunctions that just is the class of
probabilityfunctions. Dempster-Shafer
functions are primarily employed as a logic of the evidential support
for hypotheses. In that realm they are a generalization of the idea of
evidential support embodied by probabilistic inductive logic.
There is some controversy as to whether such a generalization is
useful or desirable, or whether simple probability is too
narrow to represent important evidential relationships captured by
some Dempster-Shafer functions.

There is a sense in which the other two quantitative measures of
uncertainty, possibility functions and ranking
functions, are definable in terms of formulas employing the
Dempster-Shaferfunctions. But this is not the best
way to understand them. Possibility functions (Zadeh, 1965,
1978), (Dubois and Prade, 1980, 1990) are generally read as
representing the degree of uncertainty in a claim, where such
uncertainty is often attributed to vagueness or fuzziness. These
functions are formally like probability functions and
Dempster-Shafer functions, but they subscribe to a simpler
addition rule: the degree of uncertainty of a disjunction is
the greater of the degrees of uncertainty of the
parts. Similarly, the degree of uncertainty of a conjunction
is the smaller of the uncertainties of the parts.

Ranking functions (Spohn, 1988) supply a measure of how
surprising it would be if a claim turned out to be true, rated on a
scale from 0 (not at all surprizing) to infinity. Tautologies have
rank 0 and contradictions are infinitely surprizing.
Logically equivalent claims have the same rank. The
rank of a disjunction is equal to the rank of the
lower ranking disjunct. These functions may be used to represent a
kind of order-of-magnitude reasoning about the plausibility
of various claims.

See (Halpern, 2003) for a good comparative treatment of all of these
approaches.

Here are the axioms for the Plausibility Relations and the
Qualitative Probability Relations.

Axioms for the Plausibility Relations
Each plausibility relation ⊆ satisfies the following
axioms:

if T is a tautology and K is a contradiction, it
is not the case that
T ⊆ K;

if A is logically equivalent to B and C is logically equivalent to D, and
A ⊆ C, then
B ⊆ D;

if A logically entails B, then A ⊆ B;

if A ⊆ B and B ⊆ C, then
A ⊆ C.

Two sentences are defined as equally plausible, A =
B, just when
A ⊆ B
and
B ⊆ A.
One sentence is defined as less plausible than another,
A ⊂ B,
just when
A ⊆ B but not B ⊆ A.

Axioms for the Qualitative Probability
Relations
To get the qualitative probability relations we add the
axioms

A ⊆ B or B ⊆ A;

if ‘(S and A)’ and
‘(S and B)’ are both logical
contradictions, then
A ⊆ B
just in case
(A or S) ⊆ (B or S).

The typical axioms for quantitative probability are as
follows:

for all sentences S, 0 ≤ P[S] ≤ 1;

if S is a tautology, then P[S] = 1;

if ‘(A and B)’ is a logical
contradiction, then
P[A or B]
= P[A] + P[B].

Axioms 1-6 for the qualitative probability relations are
probabilistically sound with respect to the quantitative probability
functions. That is, for each given probability function
P, define a relation ⊆ such that
A ⊆ Bjust in caseP[A] ≤ P[B].
Then ⊆ must satisfy axioms 1-6. However, not every qualitative
probability relation that satisfies axioms 1-6 may be represented by a
probability function. To get that we must add one further axiom.

Let's say that a qualitative probability relation ⊆ is
fine-grained just in case it satisfies the following
axiom:

(7) if A ⊂ B, then there is some tautology
consisting of n sentences, (S1 or
S2 or … or Sn),
where each distinct Si and
Sj are inconsistent with one another,
such that for each of the Si,
(A or Si) ⊂ B.

For each fine-grained qualitative probability relation ⊆
there is a unique probability function P such that
A ⊆ Bjust in caseP[A] ≤ P[B].

Now call a qualitative probability relation ⊆
properly extendable just in case it can be extended to a fine-grained
qualitative probability relation defined on a larger language (i.e., a language
containing additional sentences).
Then for every properly extendable qualitative probability relation
⊆ there is a probability function P such that
A ⊆ Bjust in caseP[A] ≤ P[B]. In general a given properly extendable qualitative probability relation may have many such representing probability functions, corresponding to different ways of extending it to fine-grained qualitative probability relations.

Thus, the quantitative probability functions may be viewed as just useful
ways of representing properly extendable qualitiative probability
relations on a convenient numerical scale.

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